38 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA PROBLEMS 1. Let G be a group and let 523 be a set of subsets of G, satisfying conditions (VO and (Vn) of Section 12.3, Problem 3, together with the following: (GVi) For all U e 58 there exists V 6 58 such that V - V c U. (GVn) For all U e 58 we have LT1 e 58. (GVm) For all U e 58 and a e G, we have aUa"1 e 58. Show that there exists a unique topology on G, compatible with the group structure of G, for which 83 is the set of neighborhoods of the neutral element e. 2. Every topology compatible with the group structure of a finite group G is obtained by taking as neighborhoods of the neutral element the sets containing a normal sub- group H of G. Give an example of a non-Hausdorff topological group in which the center is not closed and has as its closure a noncommutative subgroup. 3. Let G be a connected topological group. Show that every totally disconnected normal subgroup D of G is contained in the center of G. (If a e D, consider the mapping 1 of G into D.) 4. Show that the commutator subgroup of a connected topological group is connected (use (3.19.3) and (3.19.7)). 5. Let G be a topological group and let H, K be subgroups of G such that H => K => H', where H' is the commutator subgroup of H. Show that K contains the commutator subgroup of fl. Deduce that if G is Hausdorff , the closure in G of a solvable subgroup is solvable (induction on the length of the derived series). 6. Let G be a topological group and let H be a closed normal subgroup of G, containing the commutator subgroup of G. If the identity component K of H is solvable, show that the identity component L of G is solvable. (Show by using Problem 4 that K contains the commutator subgroup of L.) 7. Let 9? : R -> X be the canonical homomorphism and let 6 be an irrational number. On the topological space G = R2 x T2, a group law is defined by In this way G becomes a locally compact group (even a Lie group). Show that the commutator subgroup of G is not closed in G. 8. Let (Ga)a6i be any family of topological groups. Show that the product topology on G =0 Got is compatible with the product group structure (Section 12.5, Problem 4). «6l The topological group so defined is called the product of the topological groups G« , Let H be the normal subgroup of G consisting of all x = (xa) such that for all but a finite number of indices *« is the identity element of G« . Show that H is dense in G.osed.of the